Theorem wl-1mintru1

Description: Using the recursion formula:

"(n+1)-mintru-(m+1)" <-> if- ( ph , "n-mintru-m" , "n-mintru-(m+1)" )

for "1-mintru-1" (meaning "at least 1 out of 1 input is true") by plugging in n = 0, m = 0, and simplifying. The expressions "0-mintru-0" and "0-mintru-1" are base cases of the recursion, meaning "in a sequence of zero inputs, at least 0 / 1 input is true", respectively equvalent to T. / F. .

Negating an "n-mintru1" operation means: All n inputs ph .. th are false. This is also conveniently expressed as -. ( ph \/ .. \/ th ) . Applying this idea here (n = 1) yields the obvious result that in an input sequence of size 1 only then all will be false, if its single input is. (Contributed by Wolf Lammen, 10-May-2024)

	Ref	Expression
Assertion	wl-1mintru1	$$⊢if-\left(\chi ,\top ,\perp \right)\leftrightarrow \chi$$

Proof

Step	Hyp	Ref	Expression
1		tbtru	$$⊢\chi \leftrightarrow \left(\chi \leftrightarrow \top \right)$$
2	1	biimpi	$$⊢\chi \to \left(\chi \leftrightarrow \top \right)$$
3		nbfal	$$⊢\neg \chi \leftrightarrow \left(\chi \leftrightarrow \perp \right)$$
4	3	biimpi	$$⊢\neg \chi \to \left(\chi \leftrightarrow \perp \right)$$
5	2 4	casesifp	$$⊢\chi \leftrightarrow if-\left(\chi ,\top ,\perp \right)$$
6	5	bicomi	$$⊢if-\left(\chi ,\top ,\perp \right)\leftrightarrow \chi$$